What makes duration higher or lower

What makes a bond’s duration higher or lower?

One idea explains all three levers: duration is a weighted average of time, so anything that shifts weight toward the final payment raises it. A longer maturity pushes the final payment further out. A lower coupon leaves more of the bond’s value in that final payment. A lower yield discounts distant cash flows less, so they keep more weight. The same directions hold for modified duration.

Duration (interest rate risk)other things equal

• Higher duration Memorize more weight on distant cash flows
  • Longer time to maturity Memorize the final payment is further away
  • Lower coupon rate Memorize less cash early, more weight on the final payment
  • Lower yield-to-maturity Memorize distant flows are discounted less
• Lower duration Memorize more weight on near cash flows
  • Shorter time to maturity Memorize
  • Higher coupon rate Memorize
  • Higher yield-to-maturity Memorize
  • More of the coupon period elapsed Use it until the coupon is paid, then it jumps up

Increase in…	Effect on duration	Why
Time to maturity	Up (almost always)	The final payment is further away.
Coupon rate	Down	Bigger early cash flows carry more weight.
Yield-to-maturity	Down	Distant cash flows are discounted harder, so they weigh less.
Fraction of the coupon period elapsed	Down, until the coupon is paid, then up	Every cash flow is closer.

Here are the three levers in numbers (annual-pay bonds, Macaulay duration in years):

Lever	Bonds	Duration
coupon effect	10-year bonds at a 5% yield: coupon 2%, 5%, 8%	9.01, 8.11, 7.54
maturity effect	5% coupon bonds at a 5% yield: 5, 10, 20 years	4.55, 8.11, 13.09
yield level effect	10-year, 5% coupon bonds: yield 3%, 5%, 7%	8.27, 8.11, 7.94

Exam tip For “which bond has the highest duration”, go in this order: longest maturity, lowest coupon, lowest yield. A zero-coupon bond beats any coupon bond of the same maturity.

Two bonds with the same maturity and yield, one at a premium and one at a discount: the premium bond has the higher coupon, so it has the lower duration.

What are the durations of zero-coupon, perpetual and floating-rate bonds?

Special-case bondsthe useful extremes

• Zero-coupon bond Memorize MacDur = maturity; ModDur = T ÷ (1 + y)
• Perpetual bond Memorize MacDur = (1 + r) ÷ r; ModDur = 1 ÷ r; finite
• Floating-rate note Memorize MacDur = time left to the next reset
• Long deep-discount bond Use it duration can peak above (1 + r) ÷ r, then fall back

Zero-coupon bonds

A zero pays once, at maturity, so that single cash flow has a weight of 1. The duration of a zero-coupon bond equals its time to maturity, and its modified duration is that time divided by one plus the yield. No coupon bond of the same maturity has a higher duration.

$$\text{MacDur} = T, \qquad \text{ModDur} = \frac{T}{1 + y}$$

Perpetual bonds

A perpetual bond never matures, yet its duration is finite. Payments far into the future are discounted so heavily that they carry almost no weight. The duration of a perpetual bond depends only on the yield.

$$\text{MacDur} = \frac{1 + r}{r}, \qquad \text{ModDur} = \frac{1}{r}$$

Floating-rate notes

A floater’s coupon resets to the market reference rate at set dates. Price risk therefore exists only until the next reset, when the coupon catches up with the market. The duration of a floating-rate note is the fraction of the period left to the next reset, converted to years.

$$\text{MacDur} = \frac{T - t}{T} \text{ periods}$$

Here $T$ is the days in the reset period and $t$ the days since the last reset. Divide by the number of periods a year to annualize.

Two curiosities the exam likes

The saw-tooth pattern

At a constant yield, Macaulay duration drifts down smoothly between coupon dates and jumps up just after each coupon is paid. This is the saw-tooth pattern . Time passing brings every payment closer, but paying a coupon removes the nearest payment, which leaves a longer average wait.

Long deep-discount bonds

For a par or premium bond, duration climbs toward the perpetuity level $(1 + r)/r$ from below as maturity lengthens, and never passes it. For a long bond priced at a deep discount (coupon below the yield) the pattern differs: duration can rise above $(1 + r)/r$, peak, and then fall back toward it. Our numbers, for a 3% coupon at a 7% yield, where the perpetuity level is 1.07 ÷ 0.07 = 15.29:

Maturity	20 years	30 years	40 years	60 years	100 years
Macaulay duration	13.56	15.77	16.37	16.03	15.40

So a longer bond is not always riskier, but only in this corner case. For everything else, longer means more duration.

Go deeper Why the closed form produces the hump

The closed-form Macaulay duration is $\frac{1+r}{r}$ minus a fraction whose numerator contains $N(c - r)$. When the coupon is at or above the yield, $c - r \ge 0$ and the subtracted fraction is positive, so duration stays below $(1+r)/r$. When the coupon is below the yield and $N$ is large, that numerator can turn negative, so the “subtraction” adds, and duration exceeds the perpetuity level for a while. You need the shape and the conclusion, not the algebra.

Go deeper Floaters still carry some spread risk, and duration is not fixed

A floater’s low duration is against moves in the reference rate. If the market’s required spread for the issuer’s credit changes, its price still moves. That spread risk is measured in the credit risk module with the same arithmetic as this one.

Duration is also a property of today’s yield: as yields move, the duration itself changes. That second-order effect is convexity, the subject of the next module.